How to achieve uniform thickness with no breaks when extruding faces?
If you extrude individual faces these faces will not be connected anymore. This is why you get discontinuities.

Extrude your faces by pressing E instead (this is equal to Extrude and Move on Normal). Then immediatly press Rmb or Esc to cancel the translation. now the extruded faces should be at the same location as the previous faces.

To get walls with an even thickness use Shrink/Fatten by pressing AltS. By moving your mouse you can change the thickness of your walls. To get even thickness independent of corner angles hold down the Alt key while moving the mouse. If you are satisfied press Lmb to apply the tranformation.